Finite-embedded coordinate designed transformation-optical devices

ABSTRACT

The design method for complex electromagnetic materials is expanded from form-invariant coordinate transformations of Maxwell&#39;s equations to finite embedded coordinate transformations. Embedded transformations allow the transfer of electromagnetic field manipulations from the transformation-optical medium to another medium, thereby allowing the design of structures that are not exclusively invisible. A topological criterion for the reflectionless design of complex media is also disclosed and is illustrated in conjunction with the topological criterion to design a parallel beam shifter and a beam splitter with unconventional electromagnetic behavior.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a Continuation application of U.S. application Ser.No. 12/268,295 filed Nov. 10, 2008, now U.S. Pat. No. 8,837,031, issuedSep. 16, 2014, which claims the benefit of priority from provisionalapplication No. 60/987,014 filed Nov. 9, 2007, and from provisionalapplication No. 60/987,127 filed Nov. 12, 2007, the contents of whichare incorporated herein by reference.

FIELD

The technology herein relates to optical phenomena, and moreparticularly to complex electromagnetic materials that allow thetransfer of electromagnetic field manipulations from thetransformation-optical medium to another medium. The technology hereinalso relates to parallel beam shifters and beam splitters withunconventional electromagnetic behavior.

BACKGROUND AND SUMMARY

Metamaterials offer an enormous degree of freedom for manipulatingelectromagnetic fields, as independent and nearly arbitrary gradientscan be introduced in the components of the effective permittivity andpermeability tensors. In order to exploit such a high degree of freedom,a viable method for the well-aimed design of complex materials would bedesirable.

Pendry et al., Science 312, 1780 (2006) reported a methodology based oncontinuous form-invariant coordinate transformations of Maxwell'sequations which allows for the manipulation of electromagnetic fields ina previously unknown and unconventional fashion. This method wassuccessfully applied for the design and the experimental realization ofan invisibility cloak and generated widespread interest specifically inthe prospects of electromagnetic cloaking—a topic that has dominatedmuch of the subsequent discussion.

The methodology presented in Pendry et al makes use of form-invariantcontinuous coordinate transformations of Maxwell's equations. The use ofcontinuous transformations provides a complex transformation-opticalmaterial which is invisible to an external observer. In other words, thefield modifications precipitated in the transformation-optical devicegenerally may not be transferred to another medium and the originalelectromagnetic properties of waves impinging on the medium are restoredas soon as the waves exit the optical component. Transformation-opticaldesigns reported in the literature so far generally have in common thatthe electromagnetic properties of the incident waves are exclusivelychanged within the restricted region of the transformation-opticaldevice. However, for the sake of the continuity of the transformation,the field manipulation cannot be transferred to another medium or freespace and thus remains an, in many cases, undesired local phenomenon.

It would be desirable to have also a tool for the design ofelectromagnetic/optical components that takes advantage of the highdegree of design freedom provided by the transformation-opticalapproach, but allows the transfer of field manipulations outside thetransformation-optical material. Such a method would allow the creationof optical devices with unconventional electromagnetic/optical behaviorand functionality that exceeds the abilities of conventional componentslike lenses, beam steerers, beam shifters, beam splitters and similar.

The technology herein expands the design method for complexelectromagnetic materials from form-invariant coordinate transformationsof Maxwell's equations to finite embedded coordinate transformations. Incontrast to continuous transformations, embedded transformations allowthe transfer of electromagnetic field manipulations from thetransformation-optical medium to another medium, thereby allowing thedesign of structures that are not exclusively invisible. Theillustrative exemplary non-limiting implementations provide methods todesign such novel devices by a modified transformation-optical approach.The conceived electromagnetic/optical devices can be reflectionlessunder certain circumstances.

The exemplary illustrative non-limiting technology herein furtherdelivers a topological criterion for the reflectionless design ofcomplex media. This exemplary illustrative non-limiting expanded methodcan be illustrated in conjunction with the topological criterion toprovide an example illustrative non-limiting parallel beam shifter andbeam splitter with unconventional electromagnetic behavior.

The concept of embedded coordinate transformations significantly expandsthe idea of the transformation-optical design of metamaterials whichitself was restricted to continuous coordinate transformations so far.The expansion to embedded transformations allows for non-reversiblychange to the properties of electromagnetic waves in transformationmedia and for transmission of the changed electromagnetic properties tofree space or to a different medium in general. In order to design themedium as reflectionless, a new topological criterion for the embeddedtransformations can be used to impose constraints to the metric of thespaces at the interface between the transformation-optical medium andthe surrounding space. This metric criterion can be applied in theconception of a parallel beam shifter and a beam splitter and confirmedin 2D full wave simulations. Such exemplary illustrative non-limitingdevices can provide an extraordinary electromagnetic behavior which isnot achievable by conventional materials. Such examples clearly statethe significance of embedded coordinate transformations for the designof new electromagnetic elements with tunable, unconventional opticalproperties.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other features and advantages will be better and morecompletely understood by referring to the following detailed descriptionof exemplary non-limiting illustrative implementations in conjunctionwith the drawings of which:

FIG. 1 shows an exemplary illustrative non-limiting implementation of alinear spatial coordinate transformation for the conception of anexemplary illustrative non-limiting parallel beam shifter of thickness2d (in this example, the maximally allowed beam width of the incomingbeam is 2h1);

FIG. 2 shows an exemplary illustrative non-limiting implementation of anon-linear spatial coordinate transformation of second order for anexemplary illustrative non-limiting beam divider;

FIGS. 3 a-3 d show illustrative exemplary illustrative non-limitingelectric field phasors for a convergent wave at normal (a+b) and obliqueincidence (c+d) on a parallel beam shifter for two different shiftpositions;

FIG. 4 a-4 f shows exemplary illustrative non-limiting electric fielddistribution (color map) and power flow lines (grey) of a parallel beamshifter for diffracting plane waves with shift parameters (a) a=1:8, (b)−1.8, for a convergent beam under perpendicular incidence with (c) a=2,(d)=−2 and for oblique incidence with (e) a=2 and (f) a=−1:2;

FIGS. 5 a-5 f show illustrative non-limiting electric field distribution(color map), power flow lines (grey) (a+d) and power flow on a linear(b+e) and logarithmic scale (c+f) of a beam splitter for diffractingplane waves with shift parameters (a-c) |a|=15 for perpendicularincident waves and (d-f) |a|=12 for oblique incidence;

FIG. 6 shows an exemplary illustrative non-limiting beam focusing (a)and expanding unit (b) designed by the embedded coordinatetransformation approach;

FIG. 7 shows a diagram of an exemplary illustrative non-limiting systemfor designing and manufacturing a transformation-optical device; and

FIG. 8 shows an exemplary illustrative non-limiting fabrication process.

FIG. 9 shows another exemplary illustrative non-limiting computerimplemented process for designing and manufacturingtransformation-optical devices using embedded coordinate transformationsof electromagnetic permittivity and permeability tensors.

DETAILED DESCRIPTION

The exemplary illustrative non-limiting technology herein provides ageneralized approach to the method of form-invariant coordinatetransformations of Maxwell's equations based on finite embeddedcoordinate transformations. The use of embedded transformations adds asignificant amount of flexibility to the transformation design ofcomplex materials. For example, with finite-embedded transformations, itis possible to transfer field manipulations from thetransformation-optical medium to a second medium, eliminating therequirement that the transformation optical structure be invisible to anexternal observer. The finite-embedded transformation thus significantlybroadens the range of materials that can be designed to includedevice-type structures capable of focusing or steering electromagneticwaves. Like transformation optical devices, the finite-embeddedtransform structures can be reflectionless under conditions that wedescribe below. The general methodology is graphically illustrated belowfor a parallel beam shifter.

FIG. 1 illustrates a possible linear coordinate transformation for thisexemplary non-limiting design. Mathematically, the transformation in thegrey shadowed region I shown in FIG. 1 is described by

x′(x,y,z)=x

y′(x,y,z)=y+a(x+b)

z′(x,y,z)=z  (1-3)

As can be seen, the transformation only affects the y-coordinates whilethe x- and z-coordinates remain unchanged. The y-coordinates in FIG. 1are thus tilted at an angle φ=atan(a), where a defines the slope of theslanted y-coordinate lines. Note that the coordinate transformation inthe domains II and III are completely unrelated to the functionality ofthe beam shifter, as no electromagnetic fields penetrate into theseregions.

The material properties in regions II and III can be arbitrarily chosenand do not have to be considered in the following discussion. As pointedout in D. Schurig et al, Opt. Express 14, 9794 (2006) and E. J. Post,Formal Structure of Electromagnetics, Dover Publications (1997), thematerial properties of a transformation-optical medium can be calculatedas

ε^(i′j′)=det(A _(i) ^(i′))⁻¹ A _(i) ^(i′) A _(j) ^(j′)ε^(ij)

μ^(i′j′)=det(A _(i) ^(i′))⁻¹ A _(i) ^(i′) A _(j) ^(j′)μ^(ij)  (4-5)

for a given coordinate transformation x^(a′)(x^(a))=A_(a) ^(a′)x^(a),where A_(a) ^(a′) is the Jacobi matrix and det(A_(a) ^(a′)) itsdeterminant.

The coordinate systems at the interface at x=d between region I and freespace appear to be discontinuous. As a discontinuous coordinatetransformation can be considered as the limit of a continuous coordinatetransformation, the discontinuous transition at the boundary at x=d maybe taken into account in order to rigorously apply the same formalism asdescribed in J. B. Pendry, et al., Science 312, 1780 (2006); and D.Schurig et al, Opt. Express 14, 9794 (2006), for the continuouscoordinate transformation. In this case, the material properties at theinterface will carry the character of the discontinuity at the boundary.Mathematically, the discontinuity at the interface between region I andfree space can be described by

x′(x,y,z)=x

y′(x,y,z)=θ(d−x)[y+a(x+d)]+θ(x−d)y

z′(x,y,z)=z  (6-8)

where θ(ξ)=1 for ξ>0 and 0 for ξ<0. Calculating the permittivity andpermeability tensors by use of equations (4-5), one obtains

$\begin{matrix}{{ɛ^{ij} = {\mu^{ij} = {\frac{1}{a_{22}}\begin{pmatrix}1 & a_{12} & 0 \\a_{12} & {a_{12}^{2} + a_{22}^{2}} & 0 \\0 & 0 & 1\end{pmatrix}}}}{with}} & (9) \\{{a_{12} = {{{\theta \left( {d - x} \right)}a} - {{\delta \left( {x - d} \right)}\left\lbrack {a\left( {x + d} \right)} \right\rbrack}}}{a_{22} = {{\theta \left( {d - x} \right)} + {{\theta \left( {x - d} \right)}.}}}} & \left( {10\text{-}11} \right)\end{matrix}$

Refer to the Appendix for a more detailed mathematical analysis.

The material properties in equations (9-11) above can be interpreted intwo distinct ways. In the first case, the transformation is consideredto be discontinuous at the boundary, so that the interface has to betaken into account and the material properties at the boundary containall terms in equations (10) and (11) above. The delta distribution,being the derivative of the Heaviside function, carries the signature ofthe discontinuity at the boundary between region I and free space inFIG. 1. As in the ray approximation, the y-coordinate lines in FIG. 1represent the direction of the power flow for a beam propagating fromx=∞ . . . ∞, the inclusion of the delta distribution in the materialparameters results in the trajectory indicated by the black arrows. Theincoming wave will be shifted in the y-direction and abruptly be forcedback on its old path at the boundary at x=d, thereby rendering theentire beam shifting section invisible to an observer.

The exemplary illustrative non-limiting implementation herein providesan alternate interpretation of the calculated material parameters. Inthis context, the discontinuity at the boundary is not considered in thecalculation of the material properties. This means that the deltadistribution, which is responsible for the backshift of the beam to itsold path at x=d, is not included in equation (10). In other words, thematerial properties are calculated only within thetransformation-optical material without taking the interface to freespace into account and then embedded into free space. The linearcoordinate transformation (I=1) for the design of a parallel beamshifter illustrated in FIG. 1 thus shows the space within the greyshadowed region I(θh₁−|f₁

|)=1) is tilted at an angle φ=arctan(α) with respect to free space,which inherently results in a compression of space in region II and adilution of space in region III. Considering the boundary between regionI and free space, the coordinate systems are seen to be discontinuous atthe interface x=d.

In more detail, it is useful to explain the difference between “embeddedtransformations” and “discontinuous transformations”. Interpreting thetransformation as discontinuous, the boundary must be taken into accountand the transformation of the y-coordinate at the transition from regionI and free space must read

y¹(x,y,z)=θ(d−x)[y+ak₁(x,y)]+θ(x−d)y  (18)

so that a₂₁|_(a=d)∞δ(d−x). As in the ray approximation the y-coordinatelines in FIG. 1 represent the direction of the power flow for a beampropagating from x=−∞ to ∞. The inclusion of the delta distribution inthe material parameters would result in the trajectory indicated by theblack arrows. The incoming wave would be shifted in the y-direction andabruptly be forced back on its old path at the boundary at x=d, therebyrendering the entire beam shifting section invisible to an observer.However, as can be seen from equation (4) immediately above, theboundary is not included in the calculation of the material propertiesfor the beam shifter and beam splitter. The coordinate transformation iscarried out locally for the transformation-optical medium and thenembedded into free space which results in the trajectory indicated bythe green arrows. The beam is shifted in the y-direction and maintainsits lateral shift after exiting the transformation-optical medium.

This method is similar to the “embedded Green function” approach in thecalculation of electron transport through interfaces (see, J;. E.Inglesfield, J. Phys. C: Solid State Phys. 14, 3795 (1981), so that werefer to it as an “embedded coordinate transformation”. For this case,the beam in FIG. 1 follows the green arrows. It is shifted in thelateral y-direction and preserves its lateral shift after exiting thetransformation-optical medium. This alternative method of “embeddedcoordinate transformations” paves the way to a novel class oftransformation-optical devices which are not invisible to an observer.Instead, the field manipulations are transferred from thetransformation-optical medium to another medium.

At this point, the question arises as to which conditions must hold forthe embedded transformation in order to design a reflectionless opticaldevice. We found as a necessary—and in our investigated cases alsosufficient—topological condition for our exemplary illustrativenon-limiting implementation is that the metric in and normal to theinterface between the transformation-optical medium and thenon-transformed medium (in this case free space) must be continuous tothe surrounding space. In the case of an exemplary illustrativenon-limiting beam shifter, this means that the distances as measuredalong the x-, y- and z-axis in the transformation optical medium andfree space must be equal along the boundary (x=d). As can be clearlyseen from FIG. 1, this condition is fulfilled by the embedded coordinatetransformation of the beam shifter within the green shadowed region I.The material properties in the domains II and III can be arbitrarilychosen as no fields penetrate into these regions.

A second, more sophisticated example a beam splitter is shown in FIG. 2for the case of a nonlinear transformation of second order. The materialproperties are described by equation (16) in the Appendix with (p=2) and(I=2). This specific coordinate transformation is illustrated in FIG. 2.The underlying metric describes the gradual opening of a wedge-shapedslit in the y-direction. The metric in the x-direction is not affectedby the transformation. Similar to the parallel beam shifter, the beamsplitter obeys the topological condition in order to operate withoutreflection.

2D full-wave simulations can be carried out to adequately predict theelectromagnetic behavior of waves impinging on a beam shifter and a beamdivider, respectively. The calculation domain can be bounded byperfectly matched layers. The polarization of the plane waves can bechosen to be perpendicular to the x-y plane. Such 2D full wavesimulations can be used to confirm the propagation properties of wavesimpinging on a transformation-optical parallel beam shifter.

FIG. 3 illustrates the phasor of the electric field with polarization inthe direction perpendicular to the plane of propagation. The grey linesin FIG. 3 depict the power flow. The impinging beam is chosen to beconvergent. We investigated the cases of perpendicular (FIGS. 3 a+b) andoblique incidence (FIGS. 3 c+d). In both cases the parallel beam shiftershifts the beam into the y-direction without altering the angle of thephase fronts. As predicted, the lateral shift is maintained after thebeam exits the transformation-optical device. The parallel beam shifteris found to operate without reflection in agreement with the topologicalcriterion we considered in the design process. The parallel beam shiftershows a well-designed behavior which cannot be achieved by conventionaloptical elements as it shifts the beam without tilting the phase fronts.The presented parallel beam shifter can play a crucial role inconnection with tunable, reconfigurable metamaterials as it allowsscanning of a beam focus along a flat surface without changing the planeof the focus and without introducing a beam tilt or aberrations. Theseproperties become even more significant for applications where shortworking distances are used between scanner and object.

FIG. 4 shows an exemplary illustrative non-limiting electric fielddistribution (color map) and power flow lines (grey) of a parallel beamshifter for diffracting plane waves with shift parameters (a) a=1:8, (b)−1.8, for a convergent beam under perpendicular incidence with (c) a=2,(d)=−2 and for oblique incidence with (e) a=2 and (f) a=−1:2. FIG. 4depicts the spatial distribution of the real part of thetransverse-electric phasor (color map) and the direction of the powerflow (grey lines) of propagating waves and oblique incidence (FIG. 4e-f). The curvature of the incoming wave fronts was freely chosen to beplane (a-b) or convergent (c-f). As can be seen from FIG. 4 a-b, thebeam shifter translates the incoming plane wave in the y-directionperpendicular to the propagation x-direction without altering the angleof the wave fronts. In contrast, the direction of the power flow changesby an angle φ=arctan(α) (α: shift parameter) with respect to the powerflow of the incoming plane wave. After propagation through the complextransformation-optical medium, the wave fronts and the power flowpossess the same direction as the incoming beam, however the position ofthe wave is offset in the y-direction. The shift parameter a wasarbitrarily chosen to be 1.8 (FIG. 3 a) and −1.8 (FIG. 4 b).

A similar behavior can be observed for waves with wave fronts ofarbitrary curvature, as for example for convergent waves (FIG. 4 c-f).In this case, the focus of the beam can be shifted within a planeparallel to the y-axis by variation of the shift parameter a. As isobvious from FIG. 4 e-f, the same behavior applies for incoming waves atoblique incidence. The beam solely experiences a translation in they-direction whereas the x-position of the focus remains unchanged. Inall cases, the realized transformation-optical parallel beam shifterproves to operate without reflection confirming our metric criterionused for the design. The presented parallel beam shifter could play acrucial role in connection with tunable, reconfigurable metamaterials asit would allow scanning of a beam focus along a flat surface withoutchanging the plane of the focus and without introducing a beam tilt oraberrations. These properties become even more significant forapplications where short working distances are used between scanner andobject.

As a second example for the strength of the new design tool, atransformation-optical beam splitter was conceived. FIG. 5 shows anexample illustrative non-limiting electric field distribution (colormap), power flow lines (grey) (a+d) and power flow on a linear (b+e) andlogarithmic scale (c+f) of a beam splitter for diffracting plane waveswith shift parameters (a-c) |a|=15 for perpendicular incident waves and(d-f) |a|=12 for oblique incidence. The 2D full wave simulation resultsare illustrated in FIG. 5 a-f. FIGS. 5 a-c and 5 d-f show the spatialdistribution of the transverse electric field (a+d) and the power flowon a linear (b+e) and a logarithmic scale (c+f) for normal and obliqueincidence, respectively. The incoming beam is split into two beams witha small fraction of leaking fields in the gap due to scattering anddiffraction. The beam splitter does not show any reflection.

In more detail, FIG. 5 a shows exemplary illustrative non-limitingelectric field distribution and the power flow lines for waves atperpendicular incidence. The beam splitter shifts one half of the wavein the (+y)-direction and the second half in the (−y)-direction, thussplitting the wave at the mid-point. The split waves are not perfectlyparallel at the exit plane of the device due to diffraction of theincoming wave of finite width. As can be seen, there exists a smallfraction of scattered fields within the split region which can beexplained in terms of diffraction and scattering which is out of thescope of this letter. The beam splitter was found to operate withoutreflection in agreement with the metric criterion.

FIG. 5 b shows an exemplary illustrative non-limiting display of thenormalized power flow inside and outside the device. In order to enhancethe contrast in the visualization of the power distribution at the beamsplitter output, the color scale is saturated inside the beam splittermedium. As obvious from the transformation (FIG. 2), the power densityinside the transformation optical medium is expected to be higher thanoutside the material, which is indicated by the density of the gridlines in FIG. 2 and confirmed by the simulations. The power flow densityabruptly decreases at the output facet of the beam splitter. Forclarification, FIG. 5 c shows the power flow on a logarithmic scale. Byintegration of the power density inside the gap region between the beamsand the power density inside either the upper or lower arm of the splitbeams a power ratio of 10:1 was calculated. The scattered waves in thegap carried about 4% of the total power.

In FIG. 5 d-f, the spatial distribution of the transverse electric field(d) and the power flow on a linear (e) and a logarithmic scale (f) areshown for an obliquely incident wave on the beam splitter. Again, thebeam is clearly divided into two beams with a small fraction ofdiffracted and scattered fields inside the gap between the split beams.As for perpendicular incidence, no reflection was observed in thesimulation at both the input and output facet of the beam splitter.Similar to the result for perpendicular incidence, the propagationdirections of the outgoing waves are not parallel. In addition, theangles of the central wave vectors of the split beams with refer to thecentral wave vector of the incident beam are not equal.

A further example beam focusing and expanding unit is shown in FIGS. 6 aand 6 b. Again the design was based on embedded coordinatetransformations. Although the devices provide the aimed functionality offocusing and expanding electromagnetic waves, the devices are notreflectionless due to a metric mismatch at the right boundary betweenthe transformation-optical devices and free space. However at thispoint, it is in principle possible to add anti-reflection-coatings atthe right interface to suppress the reflections.

FIG. 7 shows a diagram of an exemplary illustrative non-limiting systemfor designing and manufacturing a transformation-optical device.Computing system or processor 100 comprises a programmable computer foraccepting optical device parameters and transformation mediumparameters. Computing system 100 uses these parameters to computetransformation coordinates for specifying dimensional metamaterialpermittivity and permeability characteristics of an optical device inaccordance with specific embedded coordinate transformations specific tothe device. Computed two-dimensional or three-dimensional metamaterialfabrication process specifications are provided to optical devicemanufacturing process equipment 200 to produce an optical device havingthe desired transformation optical 300 characteristics.

FIG. 8 shows an exemplary illustrative non-limiting computer implementedprocess for designing and manufacturing transformation-optical devicesusing embedded coordinate transformations of electromagneticpermittivity and permeability tensors. The process includes:

computing a local form-invariant coordinate transformation of Maxwell'sEquations for a transformation-optical medium of an optical device(block 302);

computing form-invariant coordinate transformations for a regionsurrounding the transformation-optical medium of the device (block 304);

embedding the local form-invariant computed coordinate transformationinto free space region or a second surrounding optical medium definingthe device (block 306); and

providing the computed coordinate transformations to a metamaterialfabrication process equipment (block 308).

FIG. 9 shows another exemplary illustrative non-limiting computerimplemented process for designing and manufacturingtransformation-optical devices using embedded coordinate transformationsof electromagnetic permittivity and permeability tensors. The processincludes:

recording initial configurations of E-M fields in a medium on aCartesian coordinate mesh (block 402);

distorting/changing the Cartesian coordinate mesh to a predeterminedconfiguration represented by a distorted coordinate mesh (block 404);

recording the distortions as a coordinate transformation between theCartesian mesh and the distorted coordinate mesh (block 406);

computing scaled E-M field values permittivity c and permeability pvalues in accordance with the coordinate transformations (block 408);

embedding the computed E-M field values into the coordinate spacedefining a transformation-optical device (block 410), wherebytransformations for the finite region of space defining atransformation-optic device are defined;

using the scaled μ and ε values to set/define the two-dimensional and/orthree-dimensional spacial permittivity and permeability properties of areconfigurable or non-reconfigurable transformation-opticalstructure/device composed of metamaterials;

and

providing the computed transformed values defining thetransformation-optical device to a metamaterial fabrication processingequipment that fabricates the metamaterial for the optical device (block412).

As should be clear from the above, all of the devices illustrated abovecan be realized in a relatively straightforward manner usingartificially structured metamaterials. Techniques for designing thesebasic elements for metamaterials are now well-known to the community.One of the advantages of using metamaterials is that tunability can beimplemented, allowing the materials to change dynamically betweenseveral transformation optical states, or continuously over a range oftransformation-optical states. In this way, a set of dynamicallyreconfigurable devices to control electromagnetic waves can be designedinto a structure that can also be used for load-bearing applications.

All publications cited above are hereby incorporated herein byreference.

While the technology herein has been described in connection withexemplary illustrative non-limiting implementations, the invention isnot to be limited by the disclosure. The invention is intended to bedefined by the claims and to cover all corresponding and equivalentarrangements whether or not specifically disclosed herein.

APPENDIX

The mathematical formalism used for the calculation of the complexmaterial properties is similar to the one reported in M. Rahm, D.Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry(2007), arXiv:0706.2452v1 and D. Schurig, J. B. Pendry, and D. R. Smith,Opt. Expr. 14, 9794 (2006). For a given coordinate transformationx^(a′)(x^(a))=A_(a) ^(a′)x^(a)=(A_(a) ^(a′): Jacobi matrix, a=1 . . .3), the electric permittivity ε^(i′j′) and the magnetic permeabilityμ^(i′j′) of the resulting material can be calculated by

ε^(i,j)=[det(A_(i) ^(i′))]⁻¹A_(i) ^(i′)A  (1)

μ^(i,j)=[det(A_(i) ^(i′))]⁻¹A_(i) ^(i′)A  (2)

where det((A_(i) ^(i′)) denotes the determinant of the Jacobi matrix.For all the transformations carried out in this letter, the mathematicalstarting point is 3-dimensional euclidian space expressed in cartesiancoordinates with isotropic permittivities and permeabilitiesε^(i,j)=ε₀δ^(i,j) and μ^(i,j)=μ₀δ^(i,j) Kronecker delta).

A possible coordinate transformation for the design of a parallel beamshifter and a beam splitter consisting of a slab with thickness 2d andheight 2h can be expressed by

x′(x,y,z)=x  (3)

x′(x,y,z)=θ(h₁−|y|)[y+ak₁(x,y)]+θ(|y|−h₁)[y+γ(y)k₁(x,y)(y−s₂(y)h)]  (4)

x′(x,y,z)=z  (5)

with

$\begin{matrix}{\left. {\Theta \text{:}\mspace{14mu} \xi}\rightarrow{\Theta (\xi)} \right.:=\left\{ \begin{matrix}1 & {\xi > 0} \\{1\text{/}2} & {\xi = 0} \\0 & {\xi < 0}\end{matrix} \right.} & (6) \\{\left. {k_{l}\text{:}\mspace{14mu} \left( {\eta,\kappa} \right)}\rightarrow{k_{l}\left( {\eta,\kappa} \right)} \right.:={{s_{p}(\kappa)}\left( {\eta + d} \right)^{i}}} & (7) \\{\left. {s_{p}\text{:}\mspace{14mu} \xi}\rightarrow{s_{p}(\xi)} \right.:=\left\{ \begin{matrix}1 & {p = 1} \\\left\{ \begin{matrix}{+ 1} & {\xi \geq 0} \\{- 1} & {\xi < 0}\end{matrix} \right. & {p = 2}\end{matrix} \right.} & (8) \\{\left. {\gamma \text{:}\mspace{14mu} \vartheta}\rightarrow{\gamma (\vartheta)} \right.:=\frac{a}{{s_{2}(\vartheta)}\left( {h_{1} - h} \right)}} & (9)\end{matrix}$

where 2h1 is the maximum allowed width of the incoming beam, adetermines the shift amount and l=1 . . . n is the order of thenonlinearity of the transformation.

The transformation equations are defined for (|x|≦d), (|y|≦h) and|z|<infinity). For the case p=1, equations (3)-(5) describe a parallelbeam shifter whereas for p=2 the equations refer to a beam splitter. TheJacobi matrix of the transformation and its determinant are

$\begin{matrix}{A_{i}^{i^{\prime}} = \begin{pmatrix}1 & 0 & 0 \\a_{21} & a_{22} & 0 \\0 & 0 & 1\end{pmatrix}} & (10) \\{{\det \left( A_{i}^{i^{\prime}} \right)} = a_{22}} & (11)\end{matrix}$

with

a₂₁=θ(h₂−|f₁′|)[lak′_(i−1)]+(f₂−|h₁′|)[lγk′_(t-1) (f₁′−s₂(y′)h)]  (12)

a₂₂=θ(h₂−|f₁′|)+θ(h₂−|f₁′|)[+γk_(t)′]  (13)

where

$\begin{matrix}{\left. {f_{1}^{\prime}\text{:}\mspace{14mu} \left( {x^{\prime},y^{\prime}} \right)}\rightarrow{f\left( {x^{\prime},y^{\prime}} \right)} \right.:={y^{\prime} - {ak}_{l}^{\prime}}} & (14) \\{{\left. {f_{2}^{\prime}\text{:}\mspace{14mu} \left( {x^{\prime},y^{\prime}} \right)}\rightarrow{f\left( {x^{\prime},y^{\prime}} \right)} \right.:=\frac{y^{\prime} + {\gamma^{\prime}k_{l}^{\prime}{s_{2}\left( y^{\prime} \right)}h}}{1 + {\gamma^{\prime}k_{l}^{\prime}}}}{k_{l}^{\prime}:={k_{l}\left( {x^{\prime},y^{\prime}} \right)}}{and}{\gamma^{\prime}:={{\gamma \left( y^{\prime} \right)}.}}} & (15)\end{matrix}$

By equations (1)-(2) immediately above, it is straightforward tocalculate the tensors of the transformed relative electric permittivityε_(i,j)=ε/ε₀ and the relative magnetic permeability μ_(p)=μ/μ₀, which inthe material representation are obtained as

$\begin{matrix}{{\varepsilon_{r}^{ij} = {\mu_{r}^{ij} = {\frac{1}{a_{22}}g^{ij}}}}{where}} & (16) \\{g^{ij} = \begin{pmatrix}1 & a_{21} & 0 \\a_{21} & {a_{21}^{2} + a_{22}^{2}} & 0 \\0 & 0 & 1\end{pmatrix}} & (17)\end{matrix}$

is the metric tensor of the coordinate transformation. At this point itshould be mentioned that only the domain with θ(h₁−|f′₁|)=≡1 has to beconsidered in the material implementation which simplifies themathematical expressions.

We claim:
 1. A process of fabricating atransformation-optical/transformation-electromagnetic device comprising:computing a local form-invariant finite coordinate transformation for atransformation-optical device; computing at least one form-invariantcoordinate transformation for a region surrounding the device; embeddingthe local form-invariant finite computed coordinate transformation intothe region surrounding the device; and fabricating the device based atleast in part on the computed coordinate transformations.
 2. The processof claim 1 wherein saidtransformation-optical/transformation-electromagnetic device comprises ametamaterial.
 3. The process of claim 1 further comprising providingnon-reversibly change to the properties of electromagnetic waves in saiddevice.
 4. The process of claim 2 further including transmitting thechanged electromagnetic properties to free space.
 5. The process ofclaim 2 further including transmitting the changed electromagneticproperties to a different medium.
 6. The process of claim 1 furtherincluding enabling transference of electromagnetic field manipulationsfrom said device to a further medium.
 7. A computer implemented methodof designing transformation-optical/transformation-electromagneticdevices using embedded coordinate transformations of electromagneticpermittivity and permeability tensors, comprising: computing a localform-invariant coordinate transformation of Maxwell's Equations for atransformation-optical medium of an optical device; computingform-invariant coordinate transformations for a region surrounding thetransformation-optical medium of the device; embedding the localform-invariant computed coordinate transformation into free space regionor a second surrounding optical medium defining the device; andproviding the computed coordinate transformations to a metamaterialfabrication process equipment.
 8. A method of implementing effectivepermittivity and permeability transformation design for complexelectromagnetic materials used intransformation-optical/transformation-electromagnetic devices,comprising: computing a local coordinate transformation for thetransformation optical medium of a device; computing a coordinatetransformation for a region surrounding the transformation-opticalmedium of a device; embedding the computed local coordinatetransformation into computed coordinate transformation for the regionsurrounding the transformation-optical medium device; whereinelectromagnetic filed manipulations from a first transformation-opticalmedium can be transferred to a second medium or free space, and whereincontinuity of the transformation design across a medium-to-mediuminterface is retained.
 9. A method of implementing effectivepermittivity and permeability transformation design for complexelectromagnetic materials used intransformation-optical/transformation-electromagnetic devices,comprising: computing a local coordinate transformation for thetransformation optical medium of a device; computing a coordinatetransformation for a region surrounding the transformation-opticalmedium of a device; and embedding the computed local coordinatetransformation into computed coordinate transformation for the regionsurrounding the transformation-optical medium device, thereby enabling atransference of electromagnetic field manipulations from a firsttransformation-optical medium to a second medium.
 10. Anelectromagnetic/photonic beam-splitter, comprising: a metamaterialfabricated by computing a local coordinate transformation, computing acoordinate transformation for a region surrounding the metamaterial,embedding the computed local coordinate transformation into computedcoordinate transformation for the region surrounding the metamaterial,and incorporating the metamaterial into the beam splitter; and means fordirecting electromagnetic radiation to the metamaterial.
 11. A parallelbeam shifter, comprising: a metamaterial fabricated by computing a localcoordinate transformation, computing a coordinate transformation for aregion surrounding the metamaterial, embedding the computed localcoordinate transformation into computed coordinate transformation forthe region surrounding the metamaterial, and incorporating themetamaterial into the beam shifter; and means for directingelectromagnetic radiation to the metamaterial.
 12. A process offabricating a transformation-optical/transformation-electromagneticdevice comprising: recording initial configurations of E-M fields in amedium on a Cartesian coordinate mesh; distorting/changing thecoordinate mesh to a predetermined configuration represented by adistorted coordinate mesh; recording the distortions as a coordinatetransformation between the Cartesian mesh and the distorted coordinatemesh; computing scaled E-M field values in accordance with thecoordinate transformations; embedding the computed values into thecoordinate space defining a transformation-optical device; andfabricating the device based at least in part on said embedded computedvalues.
 13. The process of claim 12 wherein computed E-M fields arerepresented by one or more of electric displacement field D, magneticfield intensity B, or Poynting vector S.